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CZ Gate (Controlled-Z)

The CZ gate (Controlled-Z or Controlled Phase Flip) is a fundamental two-qubit quantum gate that applies a phase flip to the target qubit only when the control qubit is in the |1⟩ state. It's essential for quantum algorithms, particularly Grover's algorithm and quantum error correction.

Mathematical Definition

The CZ gate is defined by the 4×4 matrix:

\[\text{CZ} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \end{pmatrix}\]

This can be understood as:

\[\text{CZ} = |0\rangle\langle 0| \otimes I + |1\rangle\langle 1| \otimes Z\]

Action on Basis States

  • \(\text{CZ}|00\rangle = |00\rangle\) (no change)
  • \(\text{CZ}|01\rangle = |01\rangle\) (no change)
  • \(\text{CZ}|10\rangle = |10\rangle\) (no change)
  • \(\text{CZ}|11\rangle = -|11\rangle\) (phase flip)

Quantum Circuit Representation

Control ──●──
          │
Target  ──●──

Both qubits have the same symbol (●) because the CZ gate is symmetric - swapping control and target gives the same result.

Usage in Quantum Simulator

from quantum_simulator import QuantumSimulator, QuantumCircuit
from quantum_simulator.gates import H_GATE, X_GATE, CZ_GATE
import numpy as np

# Basic CZ gate application
sim = QuantumSimulator(2)
circuit = QuantumCircuit(2)

# Prepare |11⟩ state to see phase flip
circuit.add_gate(X_GATE, [0])  # |01⟩
circuit.add_gate(X_GATE, [1])  # |11⟩
circuit.add_gate(CZ_GATE, [0, 1])  # Apply CZ

circuit.execute(sim)
print(f"CZ|11⟩ = {sim.get_state_vector()}")  # Should show phase flip

# Demonstrate phase effect with superposition
def demonstrate_cz_phase():
    """Show CZ phase effect using quantum interference."""
    sim = QuantumSimulator(2)
    circuit = QuantumCircuit(2)

    # Create Bell-like superposition
    circuit.add_gate(H_GATE, [0])  # (|0⟩ + |1⟩)/√2
    circuit.add_gate(H_GATE, [1])  # Equal superposition on both qubits

    # Apply CZ - only affects |11⟩ component
    circuit.add_gate(CZ_GATE, [0, 1])

    # Rotate back to see interference
    circuit.add_gate(H_GATE, [0])
    circuit.add_gate(H_GATE, [1])

    circuit.execute(sim)
    print(f"After CZ interference: {sim.get_state_vector()}")

demonstrate_cz_phase()

Key Properties

Symmetry

The CZ gate is symmetric under qubit exchange:

\[\text{CZ}_{ij} = \text{CZ}_{ji}\]

This means CZ(control=0, target=1) = CZ(control=1, target=0).

Self-Inverse

The CZ gate is self-inverse:

\[\text{CZ}^2 = I\]

Applying CZ twice returns to the original state.

Diagonal Matrix

CZ is a diagonal gate - it only affects phases, not probability amplitudes:

  • Preserves computational basis state populations
  • Only changes relative phases between states

Relationship to Other Gates

Controlled-Z via CRZ

CZ is equivalent to CRZ(π): 

from quantum_simulator.gates import controlled_RZ
import numpy as np

cz_equivalent = controlled_RZ(np.pi)
# This is equivalent to CZ_GATE (up to global phase)

CZ from CNOT and Hadamard

CZ can be constructed from CNOT: 

──H──●──H── = ──●──
     │         │
──H──⊕──H──   ──●──


def cz_from_cnot():
    """Construct CZ using CNOT and Hadamard gates."""
    sim = QuantumSimulator(2)
    circuit = QuantumCircuit(2)

    # Prepare test state |11⟩
    circuit.add_gate(X_GATE, [0])
    circuit.add_gate(X_GATE, [1])

    # CZ implementation using CNOT
    circuit.add_gate(H_GATE, [1])     # H on target
    circuit.add_gate(CNOT_GATE, [0, 1])  # CNOT
    circuit.add_gate(H_GATE, [1])     # H on target

    circuit.execute(sim)
    return sim.get_state_vector()

Applications in Quantum Algorithms

Grover's Algorithm

CZ gates are essential for implementing Grover's diffusion operator:

def grover_diffusion_2q():
    """Implement 2-qubit Grover diffusion operator."""
    circuit = QuantumCircuit(2)

    # Step 1: Apply H to all qubits
    circuit.add_gate(H_GATE, [0])
    circuit.add_gate(H_GATE, [1])

    # Step 2: Apply X to all qubits
    circuit.add_gate(X_GATE, [0])
    circuit.add_gate(X_GATE, [1])

    # Step 3: Apply CZ (controlled phase flip)
    circuit.add_gate(CZ_GATE, [0, 1])

    # Step 4: Apply X to all qubits (undo step 2)
    circuit.add_gate(X_GATE, [0])
    circuit.add_gate(X_GATE, [1])

    # Step 5: Apply H to all qubits (undo step 1)
    circuit.add_gate(H_GATE, [0])
    circuit.add_gate(H_GATE, [1])

    return circuit

Quantum Error Correction

CZ gates are used in stabilizer codes for phase error detection:

def phase_error_syndrome():
    """Detect phase errors using CZ gates."""
    # Implementation would use CZ for syndrome extraction
    pass

Bell State Preparation

Different Bell states using CZ:

def bell_states_with_cz():
    """Create Bell states using CZ instead of CNOT."""
    # |Φ⁺⟩ = (|00⟩ + |11⟩)/√2
    circuit1 = QuantumCircuit(2)
    circuit1.add_gate(H_GATE, [0])
    circuit1.add_gate(H_GATE, [1])
    circuit1.add_gate(CZ_GATE, [0, 1])
    circuit1.add_gate(H_GATE, [1])

    return circuit1

Properties and Identities

Commutation Relations

CZ commutes with Z operations on either qubit:

\[[\text{CZ}, Z \otimes I] = 0\]
\[[\text{CZ}, I \otimes Z] = 0\]

Conjugation by Hadamard

\[(H \otimes H) \text{CZ} (H \otimes H) = \text{CNOT}\]

This shows the deep relationship between CZ and CNOT gates.

Phase Kickback

When the target is in a Z-eigenstate, the phase kicks back to the control:

def demonstrate_phase_kickback():
    """Show phase kickback with CZ gate."""
    sim = QuantumSimulator(2)
    circuit = QuantumCircuit(2)

    # Control in superposition
    circuit.add_gate(H_GATE, [0])  # (|0⟩ + |1⟩)/√2

    # Target in |1⟩ (Z eigenstate with eigenvalue -1)
    circuit.add_gate(X_GATE, [1])

    # Apply CZ - phase kicks back to control
    circuit.add_gate(CZ_GATE, [0, 1])

    # Observe kickback via interference
    circuit.add_gate(H_GATE, [0])

    circuit.execute(sim)
    return sim.get_state_vector()

Implementation Advantages

Efficiency

  • Diagonal structure makes CZ gates often easier to implement physically
  • Symmetric nature simplifies circuit design
  • Phase-only operations preserve state magnitudes

Universality

  • CZ + single-qubit gates form a universal gate set
  • Can implement any quantum algorithm
  • Natural for many quantum error correction protocols

Circuit Symbol Variations

Standard:    ──●──    or    ──●──    or    Control ──●──
             │             │              Target  ──●──

Alternative: ──CZ──         ──Z──
             ──CZ──         ──●──

Physical Implementation

CZ gates are often native operations in many quantum computing platforms:

  • Superconducting qubits: Natural interaction via tunable coupling
  • Trapped ions: Implemented via Mølmer-Sørensen gates
  • Photonic systems: Via linear optical elements

The CZ gate's diagonal nature and symmetry make it fundamental for quantum computing, providing clean phase relationships essential for quantum interference and algorithm implementation.